On Mann implicit composite subgradient extragradient methods for general systems of variational inequalities with hierarchical variational inequality constraints

• Published in: Journal of inequalities and applications Vol. 2022; no. 1; pp. 1 - 28
• Main Authors: Ceng, Lu-Chuan, Yao, Jen-Chih, Shehu, Yekini
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 10.06.2022; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Analysis; Applications of Mathematics; Approximation; Asymptotically nonexpansive mapping; General system of variational inequalities; Hilbert space; Inequalities; Lipschitzian pseudocontractive mapping; Mann implicit composite subgradient extragradient method; Mathematics; Mathematics and Statistics; Search process; Variational inequality problem; 47H10; Mann implicit composite subgradient extragradient method; 47J20; Asymptotically nonexpansive mapping; Lipschitzian pseudocontractive mapping; General system of variational inequalities; Variational inequality problem; 47H09; 47J25
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-022-02813-0

In a real Hilbert space, let the VIP, GSVI, HVI, and CFPP denote a variational inequality problem, a general system of variational inequalities, a hierarchical variational inequality, and a common fixed-point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping, respectively. We design two Mann implicit composite subgradient extragradient algorithms with line-search process for finding a common solution of the CFPP, GSVI, and VIP. The suggested algorithms are based on the Mann implicit iteration method, subgradient extragradient method with line-search process, and viscosity approximation method. Under mild assumptions, we prove the strong convergence of the suggested algorithms to a common solution of the CFPP, GSVI, and VIP, which solves a certain HVI defined on their common solutions set.